MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 2015 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let V be an n-dimensional vector space over R. From linear algebra we know that the space of all linear functionals on V forms a vector space itself, called the dual of V , i.e. ∗ V = Hom(V ; R) = ff : V ! R linearg: ∗ Elements of V are called vectors while elements of V are called covectors. If fe1; e2; ··· ; eng is a basis ∗ ∗ ∗ ∗ for V , then there is a corresponding dual basis fe1; e2; ··· ; eng for V defined by the relationship 1 when i = j; e∗(e ) = δ = i j ij 0 when i 6= j: Any v 2 V can be written uniquely as a linear combination v = a1e1 + ··· + anen, and it can be written as a column vector: 0 1 a1 B a2 C v = B C : B . C @ . A an ∗ ∗ ∗ ∗ On the other hand, any covector v 2 V can be written uniquely as v = α1e1 + ··· + αnen. It can be expressed as a row vector: ∗ v = α1 α2 ··· αn : The vector space V and its dual V ∗ have a natural (non-degenerate) pairing between them: 0 1 a1 B a2 C v∗(v) = α α ··· α B C = α a + α a + ··· + α a : 1 2 n B . C 1 1 2 2 n n @ . A an One important thing we learned from linear algebra is that although V and V ∗ are isomorphic vector spaces, they are not canonically isomorphic. In other words, there is no default isomorphism between V and V ∗. Another example of this kind is that any n-dimensional real vector space V is isomorphic to n ∼ n R by taking a basis. However, the isomorphism V = R depends on the choice of the basis. To obtain a canonical isomorphism between V and V ∗, one needs extra structure. For example, if (V; h·; ·i) is an inner product space, then the linear map ] : V −! V ∗ defined by v 7! v] := hv; ·i 1 is an isomorphism which is independent of the choice of a basis (but depends on the inner product ] ] h·; ·i). Note that if fe1; ··· ; eng is an orthonormal basis for (V; h·; ·i), then fe1; ··· ; eng is the dual basis for V ∗. One also induces an inner product on V ∗ by requiring that ] preserves their inner products. Now, we define the tensor product of two real vector spaces V and W (not necessarily of the same dimension). The tensor product V ⊗ W is the set of all formal linear combinations of v ⊗ w, v 2 V and w 2 W , with an equivalence relation ∼ identifying elements through a bilinear relation: V ⊗ W = spanfv ⊗ w : v 2 V; w 2 W g= ∼; where for any c1; c2 2 R, (c1v1 + c2v2) ⊗ w = c1(v1 ⊗ w) + c2(v2 ⊗ w); v ⊗ (c1w1 + c2w2) = c1(v ⊗ w1) + c2(v ⊗ w2): ⊗0 ⊗1 ⊗2 By convention, we set V = R. It is trivial that V = V . To see what V is, if we take a basis Pn Pn fe1; ··· ; eng, then we can write v = i=1 aiei, w = j=1 bjej, using the bilinear relations: n ! 0 n 1 n X X X v ⊗ w = aiei ⊗ @ bjejA = aibj(ei ⊗ ej): i=1 j=1 i;j=1 For example, (e1 + e2) ⊗ (e1 − e2) = e1 ⊗ e1 + e2 ⊗ e1 − e1 ⊗ e2 − e2 ⊗ e2. Note that e1 ⊗ e2 6= e2 ⊗ e1. In general, if fe1; ··· ; eng and ff1; ··· ; fmg are bases for V and W respectively, then fei ⊗ fjg forms a basis for V ⊗ W . One can define higher tensor powers V ⊗k similarly. On the other hand, we have V ∗ ⊗ W =∼ Hom(V; W ): One can identify an element v∗ ⊗ w 2 V ∗ ⊗ W with a linear map V ! W by (v∗ ⊗ w)(v) := v∗(v)w; and then extend to the rest of V ∗ ⊗W by linearity. (Exercise: Show that this is indeed an isomorphism.) Next, we define another operator called the wedge product ^. It shares similar multilinear properties but has the extra characteristic of being anti-symmetric. Let V be a real vector space as before. We define the k-th wedge product of V to be k Λ V = spanfv1 ^ v2 ^ · · · ^ vk : vi 2 V g= ∼; where for any constant c 2 R, v1 ^ · · · ^ (cvi + w) ^ · · · ^ vk = c(v1 ^ · · · ^ vi ^ · · · ^ vk) + (v1 ^ · · · ^ w ^ · · · ^ vk); v1 ^ · · · ^ vi ^ vi+1 ^ · · · ^ vk = −v1 ^ · · · ^ vi+1 ^ vi ^ · · · ^ vk: 2 These two properties imply that the wedge product is linear in each variable and that v1 ^ · · · ^ vk = 0 whenever vi = vj for some i 6= j. If fe1; ··· ; eng is a basis for V , then it is easy to see that fei1 ^ · · · ^ eik : 1 ≤ i1 < i2 < ··· < ik ≤ ng k k n forms a basis for Λ V . Hence, dim Λ V = k where n =dimV . 0 1 2 By convention, Λ V = R. Also, Λ V = V . For higher wedge products, for example, when V = R 2 k 3 with standard basis fe1; e2g, then Λ V = spanfe1 ^ e2g and Λ V = 0 for all k ≥ 3. When V = R , we have 1 Λ V = spanfe1; e2; e3g; 2 Λ V = spanfe1 ^ e2; e1 ^ e3; e2 ^ e3g; 3 Λ V = spanfe1 ^ e2 ^ e3g: Note that ΛkV = 0 whenever k >dimV (Exercise: Prove this.) The anti-symmetric property of wedge product is related to the determinant function when k =dimV . Pn Proposition 1. Let fe1; ··· ; eng be a basis for V . For any vi = i;j=1 aijej for i = 1; ··· ; n, we have v1 ^ · · · ^ vn = det(aij) e1 ^ · · · ^ en: Proof. Exercise to the reader. One can define the wedge product ΛkV ∗ for the dual space V ∗, and in fact we have a canonical isomorphism ΛkV ∗ =∼ (ΛkV )∗ defined by 0 1 f1(v1) ··· f1(vn) B . .. C (f1 ^ · · · ^ fn)(v1 ^ · · · ^ vn) = det @ . A ; fn(v1) ··· fn(vn) and extending to all the elements of ΛkV and ΛkV ∗ by linearity. For example, 1 −1 2 1 ( 2 3 ^ −1 2 ) ^ = det = 7: 0 1 −1 3 An orientation of V is a choice of [η] where 0 6= η 2 ΛnV ∗ and η ∼ cη for c > 0. For example, n ∗ ∗ ∗ n ∗ ∼ the standard orientation on R is given by [e1 ^ e2 ^ · · · ^ en]. Note that since Λ V = R, so there are only two distinct orientations on V . An ordered basis fv1; ··· ; vng for V is said to be positive if ∗ ∗ 2 η(v1 ^ · · · ^ vn) > 0. For example, if we take the standard orientation η = e1 ^ e2 of R , then the ordered basis fe1; e2g is positive while fe2; e1g is negative. Another useful operation is called contraction. Let v 2 V and ! 2 ΛkV ∗, then the contraction of ! k−1 ∗ with v, denoted by ιv! 2 Λ V is defined by (ιv!)(v1 ^ · · · ^ vk−1) = !(v ^ v1 ^ · · · ^ vk−1): 3 we have the following explicit formula for contraction: k ∗ ∗ X `−1 ∗ ∗ ∗ ∗ ιv(v1 ^ · · · ^ vk) = (−1) v` (v)(v1 ^ · · · ^ vb` ^ · · · ^ vk): `=1 ∗ ∗ ∗ ∗ For example, if v = ae1 + be2, then ιv(e1 ^ e2) = ae2 − be1. Hence, if we fix v 2 V , then we have a sequence of contractions which reduces the degree of the wedge products: 0 ι−v Λ0V ∗ ι−v Λ1V ∗ ι−·v · · ι−v ΛnV ∗: n Differential Forms in R n In this section we introduce the language of differential forms in R . Let U be an open domain in n R equipped with the standard Euclidean metric h·; ·i. The tangent space at each p 2 U is denoted by TpU, which is an n-dimensional real vector space. Note that TpU is equipped with the inner product h·; ·i so (TpU; h·; ·i) is in fact an inner product space. The tangent bundle is simply the disjoint union of all these tangent spaces: a n TU := TpU ' U × R : p2U n 1 n If we denote the standard coordinates on R by x ; ··· ; x , then we obtain an orthonormal basis f @ ; ··· ; @ g of T U. Because of this standard basis globally defined on U, we have a \canonical" @xi p @xn p p n identification of TU with U × R once a coordinate system is fixed. A vector field X on U is just n X @ X = f ; i @xi i=1 where fi : U ! R are smooth functions, ∗ Now, for each tangent space TpU, we can take the dual vector space Tp U, which is called the cotangent space at p. As for the tangent bundle TU, we can put all the cotangent spaces together to form the cotangent bundle: ∗ a ∗ n T U = Tp U ' U × R : p2U We will denote the dual basis of f @ ; ··· ; @ g in T ∗U as fdx1j ; ··· dxnj g. We can take the `-th @xi p @xn p p p p wedge product of the cotangent spaces at each point as in the previous section and then put them all together to get the bundle Λ`T ∗U. A differential `-form is just a section of Λ`T ∗U which can be expressed as X i1 ik ! = fi1···i` dx ^ · · · ^ dx ; 1≤i1<···<i`≤n for some smooth functions fi1···i` : U ! R. 4 For example, a 0-form is simply a smooth function f : U ! R. When n = 2, the 1-forms have the form ! = f(x1; x2)dx1 + g(x1; x2)dx2; and a 2-form would have the form ! = f(x1; x2)dx1 ^ dx2: We will use Ω`(U) to denote the space of all `-forms on U. Note that by the properties of wedge ` n 1 n product, we have Ω (U) = 0 whenever Ω ⊂ R with ` > n.